= Solution
A <singular system of a compact operator> $K:U\to V$ consists of positive <singular values> $\sigma_j$ and orthonormal systems $u_j\in U$, $v_j\in V$ satisfying
$$
Ku_j=\sigma_jv_j,\qquad K^*v_j=\sigma_ju_j.
$$
They span $\mathcal N(K)^\perp$ and $\overline{\mathcal R(K)}$, respectively. The <singular value decomposition> is
$$
\boxed{Ku=\sum_j\sigma_j\langle u,u_j\rangle v_j.}
$$
For an infinite-rank <compact operator>, the singular values tend to zero; for finite rank the sum is finite. Vectors in $\mathcal N(K)$ contribute nothing. This convention labels the input singular vectors by $u_j$ and the output ones by $v_j$.
For $(Ku)_j=u_j/j$, truncating after $n$ diagonal entries gives finite-rank operators with operator-norm error $1/(n+1)$, proving compactness. Its <SVD> is
$$
\boxed{u_j=v_j=e_j,\qquad\sigma_j=1/j.}
$$
Back to article page
